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Mario Abundo

Publications and source records attributed to Mario Abundo.

16 recordsLinked to original sources
arXiv2025

Optimization in the first-passage problem of a diffusion with Poissonian resetting

We address the problem of minimizing the expected first-passage time of a Brownian motion with Poissonian resetting, with respect to the resetting rate $r.$ We consider both the one-boundary and the two-boundary cases.We investigate the first-passage time (FPT) and first-exit time (FET) of a one-dimensional, time-homogeneous diffusion process subject to Poissonian resetting. We first derive a general analytical relationship that expresses the Laplace transform (LT) and the expected value of the FPT (and FET) for the process with resetting in terms of the LT of the FPT (and FET) of the underlying diffusion without resetting. This framework is then applied to determine the optimal resetting rate $r$ that minimizes the expected FPT (and FET). We provide explicit results for drifted Brownian motion and Ornstein-Uhlenbeck (OU) process. For Brownian motion, we extend existing literature by considering the case where the initial position $x$ differs from the resetting position $x_R$, providing a comprehensive parametric analysis. For the OU process, we provide new insights into the minimization of the expected FPT. Our results demonstrate how a strategic choice of the resetting rate can effectively regularize and accelerate the passage through one or two boundaries.

math.PR↗
arXiv2025

Study of direct and inverse first-exit problems for drifted Brownian motion with Poissonian resetting

\noindent We address some direct and inverse problems, for the first-exit time (FET) $\tau $ of a drifted Brownian motion with Poissonian resetting ${\cal X}(t)$ from an interval $(0,b)$ and the first-exit area (FEA) $A,$ namely the area swept out by ${\cal X}(t)$ till the time $\tau $; this type of diffusion process ${\cal X}(t)$ is characterized by the fact that a reset to the position $x_R $ can occur according to a homogeneous Poisson process with rate $r>0.$ When the initial position ${\cal X}(0)= \eta \in (0,b)$ is deterministic and fixed, the direct FET problem consists in investigating the statistical properties of the FET $\tau ,$ whilst the direct FEA problem studies the probability distribution of the FEA $A$. The inverse FET problem regards the case when $\eta $ is randomly distributed in $(0,b)$ (while $r$ and $x_R $ are fixed); if $F(t)$ is a given distribution function on the time $t$ axis, the inverse FET problem consists in finding the density $g$ of $\eta,$ if it exists, such that $P[\tau \le t ] = F(t), \ t >0.$ %In addition to the case of random initial position $\eta,$ we also study the case when the initial position $\eta$ and the resetting rate $r$ are fixed, whereas the reset position $x_R$ is random. Several explicit examples of solutions to the inverse FET problem are provided.

math.PR↗
arXiv2024

Inverse first-passage problems of a diffusion with resetting

We address some inverse problems for the first-passage place and the first-passage time of a one-dimensional diffusion process $\mathcal X(t)$ with stochastic resetting, starting from an initial position $\mathcal X(0)= \eta ;$ this type of diffusion $\mathcal X(t)$ is characterized by the fact that a reset to the position $x_R $ can occur according to a homogeneous Poisson process with rate $r>0.$ As regards the inverse first-passage place problem, for random $\eta \in (0,b), \ b < + \infty$ (and fixed $r$ and $x_R \in (0,b))$, let $\tau_{0,b}$ be the first time at which $\mathcal X(t)$ exits the interval $(0,b),$ and $\pi _0 = P(\mathcal X(\tau_{0,b}) = 0)$ the probability of exit from the left end of $(0,b);$ given a probability $q \in (0,1),$ the inverse first-passage place problem consists in finding the density $g$ of $\eta ,$ if it exists, such that $\pi _0 = q.$ Concerning the inverse first-passage time problem, for random $\eta \in (0, + \infty)$ (and fixed $r$ and $x_R >0)$, let $\tau$ be the first-passage time of $\mathcal X(t)$ through zero; for a given distribution function $F(t)$ on the positive real axis, the inverse first-passage time problem consists in finding the density $g$ of $\eta,$ if it exists, such that $P(\tau \le t ) = F(t), \ t >0.$ In addition to the case of random initial position $\eta,$ we also study the case when the initial position $\eta$ and the resetting rate $r$ are fixed, whereas the reset position $x_R$ is random. For all types of inverse problems considered, several explicit examples of solutions are reported.

math.PR↗
arXiv2023

The first-passage area of Wiener process with stochastic resetting

For a one-dimensional Wiener process with stochastic resetting ${\cal X}(t)$, obtained from an underlying Wiener process $X(t),$ we study the statistical properties of its first-passage time through zero, when starting from $x>0,$ and its first-passage area, that is the random area enclosed between the time axis and the path of the process ${\cal X} (t)$ up to the first-passage time through zero. By making use of solutions of certain associated ODEs, we are able to find explicit expressions for the Laplace transforms of the first-passage time and the first-passage area, and their single and joint moments.

math.PR↗
arXiv2021

Some examples of solutions to an inverse problem for the first-passage place of a jump-diffusion process

We report some additional examples of explicit solutions to an inverse first-passage place problem for one-dimensional diffusions with jumps, introduced in a previous paper. If $X(t)$ is a one-dimensional diffusion with jumps, starting from a random position $\eta \in [a,b],$ let be $\tau_{a,b}$ the time at which $X(t)$ first exits the interval $(a,b),$ and $\pi _a = P(X(\tau_{a,b}) \le a)$ the probability of exit from the left of $(a,b).$ Given a probability $q \in (0,1),$ the problem consists in finding the density $g$ of $\eta$ (if it exists) such that $\pi _a = q;$ it can be seen as a problem of optimization.

math.PR↗
arXiv2019

On the Fractional Riemann-Liouville Integral of Gauss-Markov processes and applications

We investigate the stochastic processes obtained as the fractional Riemann-Liouville integral of order $\alpha \in (0,1)$ of Gauss-Markov processes. The general expressions of the mean, variance and covariance functions are given. Due to the central rule, for the fractional integral of standard Brownian motion and of the non-stationary/stationary Ornstein-Uhlenbeck processes, the covariance functions are carried out in closed-form. In order to clarify how the fractional order parameter $\alpha$ affects these functions, their numerical evaluations are shown and compared also with those of the corresponding processes obtained by ordinary Riemann integral. The results are useful for fractional neuronal models with long range memory dynamics and involving correlated input processes. The simulation of these fractional integrated processes can be performed starting from the obtained covariance functions. A suitable neuronal model is proposed. Graphical comparisons are provided and discussed.

math.PR↗
arXiv2018

On the the successive passage times of certain one-dimensional diffusions

We show in detail some results, outlined in a previous paper regarding the case of Brownian motion (BM), about the distribution of the $n$th-passage time of a one-dimensional diffusion obtained by a space or time transformation of BM, through a constant barrier $a.$ Some explicit examples are reported.

math.PR↗
arXiv2018

On the first-passage area of a L$\acute{\text{e}}$vy process

Let be $X(t)= x - \mu t + \sigma B_t - N_t$ a L$\acute{\text{e}}$vy process starting from $x >0,$ where $ \mu \ge 0, \ \sigma \ge 0, \ B_t$ is a standard BM, and $N_t$ is a homogeneous Poisson process with intensity $ \theta >0,$ starting from zero. We study the joint distribution of the first-passage time below zero, $\tau (x),$ and the first-passage area, $A(x),$ swept out by $X$ till the time $\tau (x).$ In particular, we establish differential-difference equations with outer conditions for the Laplace transforms of $\tau(x)$ and $A(x),$ and for their joint moments. In a special case $(\mu = \sigma =0),$ we show an algorithm to find recursively the moments $E[\tau(x)^m A(x)^n],$ for any integers $m$ and $n;$ moreover, we obtain the expected value of the time average of $X$ till the time $\tau(x).$

math.PR↗
arXiv2016

On the joint distribution of first-passage time and first-passage area of drifted Brownian motion

For drifted Brownian motion $X(t)= x - \mu t + B_t \ (\mu >0)$ starting from $x>0,$ we study the joint distribution of the first-passage time below zero, $\tau(x),$ and the first-passage area, $A(x),$ swept out by $X$ till the time $\tau(x).$ In particular, we establish differential equations with boundary conditions for the joint moments $E[\tau(x)^m A(x)^n],$ and we present an algorithm to find recursively them, for any $m$ and $n.$ Finally, the expected value of the time average of $X$ till the time $\tau(x)$ is obtained.

math.PR↗
arXiv2015

A randomized first-passage problem for drifted Brownian motion subject to hold and jump from a boundary

We study an inverse first-passage-time problem for Wiener process $X(t)$ subject to hold and jump from a boundary $c.$ Let be given a threshold $S>X(0) \ge c,$ and a distribution function $F$ on $[0, + \infty ).$ The problem consists in finding the distribution of the holding time at $c$ and the distribution of jumps from $c,$ so that the first-passage time of $X(t)$ through $S$ has distribution $F.$

math.PR↗
arXiv2015

On the first-passage time of an integrated Gauss-Markov process

It is considered the integrated process $X(t)= x + \int _0^t Y(s) ds ,$ where $Y(t)$ is a Gauss-Markov process starting from $y.$ The first-passage time (FPT) of $X$ through a constant boundary and the first-exit time of $X$ from an interval $(a,b)$ are investigated, generalizing some results on FPT of integrated Brownian motion. An essential role is played by a useful representation of $X,$ in terms of Brownian motion which allows to reduces the FPT of $X$ to that of a time-changed Brownian motion. Some explicit examples are reported; when theoretical calculation is not available, the quantities of interest are estimated by numerical computation.

math.PR↗
arXiv2014

One-dimensional reflected diffusions with two boundaries and an inverse first-hitting problem

We study an inverse first-hitting problem for a one-dimensional, time-homogeneous diffusion $X(t)$ reflected between two boundaries $a$ and $b,$ which starts from a random position $\eta.$ Let $a \le S \le b$ be a given threshold, such that $P( \eta \in [a,S])=1,$ and $F$ an assigned distribution function. The problem consists of finding the distribution of $\eta$ such that the first-hitting time of $X$ to $S$ has distribution $F.$ This is a generalization of the analogous problem for ordinary diffusions, i.e. without reflecting, previously considered by the author.

math.PR↗
arXiv2012

The first-crossing area of a diffusion process with jumps over a constant barrier

For a given barrier $S$ and a one-dimensional jump-diffusion process $X(t),$ starting from $x<S,$ we study the probability distribution of the integral $A_S(x)= \int_0 ^ {\tau_S(x)}X(t) \ dt$ determined by $X(t)$ till its first-crossing time $\tau_S(x)$ over $S.$ In particular, we show that the Laplace transform and the moments of $A_S(x)$ are solutions to certain partial differential-difference equations with outer conditions. The distribution of the minimum of $X(t)$ in $[0, \tau_S(x)]$ is also studied. Thus, we extend the results of a previous paper by the author, concerning the area swept out by $X(t)$ till its first-passage below zero. Some explicit examples are reported, regarding diffusions with and without jumps.

math.PR↗